How to divide a circle into a square grid?
This is how I do it:



Create a circle and put it a little over the grid plane;
In Object mode select the circle first, then add the plane to the selection, by pressing &#8679; Shift RMB (the order is important);
Go to top view (Numpad 7) in Orthographic View (toggle with Numpad 5);
Go in Edit mode with &#8633; Tab;
Be sure to have all the meshes selected (hit A);
In the T panel (or just by searching with Space) find Knife project and click it.
Invert the selection with Ctrl+I
Delete the selected mesh parts with X.


Like LukeD said in another answer, this causes your mesh to have a mix of ngons, quads and tris, but that's what you asked for in your question. :-)